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The modular group $\Gamma_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a homomorphism $\Gamma_{g}\to Sp(2g)$. We also know that this homomorphism is surjective. Therefore, to understand $H^1(S,\mathbb{Q})$ as a representation of the modular group it suffices to view it as a representation of $Sp(2g)$.

Has this representation been studied somewhere? E.g. is the decomposition in irreducible representations known? I could not find this anywhere, so I apologize if this is all well-known stuff, but it is rather far from what I usually deal with. I would really appreciate a reference for these things.

Also, is it known if the map $\Gamma_{g}\to Sp(2g)$ is surjective when we restrict it to the hyperelliptic modular group? If yes, can we at least in that case understand the representation?

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  • $\begingroup$ I don't really understand what you're asking. Isn't the characterization of $H_1(S, \mathbb{Q})$ as a representation of $\mathrm{Sp}(2g)$ tautological? As in, we are realizing $\mathrm{Sp}(2g)$ as the group of automorphisms of $H_1(S, \mathbb{Q})$ which preserve the intersection pairing. $\endgroup$
    – Jeff Yelton
    Commented Feb 1, 2018 at 14:23
  • $\begingroup$ Maybe the real questions concern the homomorphism $\Gamma_g \to \mathrm{Sp}(2g)$ itself. A lot is already understood about the restriction to the hyperelliptic modular group (e.g. the precise image in $\mathrm{Sp}(2g)$), and if you ask a more specific question about this restriction I may be able to answer it. $\endgroup$
    – Jeff Yelton
    Commented Feb 1, 2018 at 14:25
  • $\begingroup$ I am sorry, I was quite confused about that. $\endgroup$
    – Klint
    Commented Feb 8, 2018 at 21:29

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For your last question: the map from the hyperelliptic modular group to $\operatorname{Sp}(2g,\mathbb{Z}) $ is not surjective as soon as $g\geq 3$. This was proved by V. Arnold, A remark on the branching of hyperelliptic integrals as functions of the parameters, Functional Anal. Appl. 2 (1968), 187–189.

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